"(3,4) Ising minimal model CFT"의 두 판 사이의 차이

2011년 6월 17일 (금) 08:44 판

introduction

Ising model

first review the minimal models page
Weber functions and conformal field theory
rank 1 case

Ising model as a minimal model

Ising model is a unitary minimal model and thus can be understood by the representation of Viraroso algebra
the representation is given by following data
\(m= 3\)
central charge \(c = 1-{6\over m(m+1)} = \frac{1}{2}\)
\(h_{p,q}(c) = {(4p-3q)^2-1 \over 48}\)
\(p= 1,2\), \(q = 1,\cdots p\)
\((p,q)=(1,1), (2,1), (2,2)\)
possible values of \(h\)
\(0, 1/2, 1/16\)

graded dimensions

associated chiral algebra has three irreducible modules with the following graded dimensions
\(\chi_0=q^{-1/48}(1+q^2+q^3+2q^4+2q^5+3q^6+\cdots)\)
\(\chi_{\epsilon}=q^{23/48}(1+q+q^2+q^3+2q^4+2q^5+3q^6+\cdots)\)
\(\chi_{\sigma}=\chi _{1,2}^{(3,4)}q^{1/24}(1+q+q^2+2q^3+2q^4+3q^5+4q^6+\cdots)\)
Rocha-Caridi character[RC84] bosonic characters of Virasoro minimal models(Rocha-Caridi formula)

\(\chi_{\sigma}=\chi _{1,2}^{(3,4)}=\chi _{2,2}^{(3,4)}=\frac{\eta (2\tau )}{\eta (\tau )}=q^{1/24}\sum _{m=-\infty }^{\infty } (-1)^mq^{3 m^2- m }=q^{1/24}\sum_{n\geq 0}\frac{q^{n(n+1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)}\)

\(\mathfrak{f}_2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})=\sqrt{2}q^{1/24}\sum_{n\geq 0}\frac{q^{n(n+1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)}\)

modularity of graded dimensions

\(\chi_M(-1/\tau)=\sum_{N} S_{M,N}\chi_N(\tau)\)

\(\chi_M(\tau+1)=\sum_{N} T_{M,N}\chi_N(\tau)\)

\(T=\left(\begin{array}{ccc}e^{-\pi i/24} & 0 & 0 \\ 0 & e^{23\pi i/24} & 0 \\ 0 & 0 & e^{\pi i/12}\end{array} \right)\)

\(2S=\left(\begin{array}{ccc}1 & 1& \sqrt{2} \\ 1 & 1 & -\sqrt{2} \\ \sqrt{2} & -\sqrt{2} & 0\end{array} \right)\)

matching two sets of funtions

http://www.research.att.com/~njas/sequences/A000009

\(f(\tau)=\frac{e^{-\frac{\pi i}{24}}\eta(\frac{\tau+1}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1+q^{n-\frac{1}{2}})\)

\(\chi_0+\chi_{\epsilon}\)

{1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4}

http://www.research.att.com/~njas/sequences/A027349

\(f_1(\tau)=\frac{\eta(\frac{\tau}{2})}{\eta(\tau)}=q^{-1/48} \prod_{n=1}^{\infty} (1-q^{n-\frac{1}{2}})\)

\(\chi_0-\chi_{\epsilon}\)

{1, -1, 0, -1, 1, -1, 1, -1, 2, -2, 2, -2, 3, -3, 3, -4, 5, -5, 5, -6, 7, -8, 8, -9, 11, -12, 12, -14, 16, -17, 18}

http://www.research.att.com/~njas/sequences/A081362

\(\chi_0=q^{-1/48}(1+q^2+q^3+2q^4+2q^5+3q^6+\cdots)\)

\(\chi_{\epsilon}=q^{23/48}(1+q+q^2+q^3+2q^4+2q^5+3q^6+\cdots)\)

{1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 3, 4}

{1, -1, 0, -1, 1, -1, 1, -1, 2, -2, 2, -2, 3, -3, 3, -4, 5, -5, 5, -6, 7, -8, 8, -9, 11, -12, 12, -14, 16, -17, 18}

sum http://www.wolframalpha.com/input/?i=(1,+0,+0,+1,+0,+1,+0,+1,+1,+1)%2B(1,+-1,+0,+-1,+1,+-1,+1,+-1,+2,+-2)

{2,-1,0,0,1,0,1,0,3,...} -> \(q^{-1/48}}(1-1/2q^{1/2}+q^{4/2}+q^{6/2}+3q^{8/2}+\cdots)\)

difference http://www.wolframalpha.com/input/?i=(1,+0,+0,+1,+0,+1,+0,+1,+1,+1)-(1,+-1,+0,+-1,+1,+-1,+1,+-1,+2,+-2)

{0,1,0,2,-1,2,-1,2,-1,...} -> \(\frac{1}{2}q^{-1/48}}(q^{1/2}+2q^{3/2}-q^{4/2}+2q^{5/2}-q^{6/2} +2q^{7/2} -q^{8/2}\cdots)\)

\(f_2(\tau)=\sqrt{2}\frac{\eta(2\tau)}{\eta(\tau)}=\sqrt{2}q^{1/24} \prod_{n=1}^{\infty} (1+q^{n})\)

\(\chi_{\sigma}=q^{1/24}(1+q+q^2+2q^3+2q^4+3q^5+4q^6+\cdots)\)

{1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89, 104, 122, 142, 165, 192, 222, 256, 296}

http://www.research.att.com/~njas/sequences/A000009

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encyclopedia

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Princeton companion to mathematics(Companion_to_Mathematics.pdf)

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@@ 25번째 줄: / 25번째 줄: @@
 <h5 style="margin: 0px; line-height: 2em; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">graded dimensions</h5>
-*  associated chiral algebra has three irreducible modules with the following graded dimensions<br><math>\chi_0=q^{-1/48}(1+q^2+q^3+2q^4+2q^5+3q^6+\cdots)</math><br><math>\chi_{\epsilon}=q^{23/48}(1+q+q^2+q^3+2q^4+2q^5+3q^6+\cdots)</math><br><math>\chi_{\sigma}=q^{1/24}(1+q+q^2+2q^3+2q^4+3q^5+4q^6+\cdots)</math><br>
+*  associated chiral algebra has three irreducible modules with the following graded dimensions<br><math>\chi_0=q^{-1/48}(1+q^2+q^3+2q^4+2q^5+3q^6+\cdots)</math><br><math>\chi_{\epsilon}=q^{23/48}(1+q+q^2+q^3+2q^4+2q^5+3q^6+\cdots)</math><br><math>\chi_{\sigma}=\chi _{1,2}^{(3,4)}q^{1/24}(1+q+q^2+2q^3+2q^4+3q^5+4q^6+\cdots)</math><br>
 *  Rocha-Caridi character[RC84] [[bosonic characters of Virasoro minimal models(Rocha-Caridi formula)]]<br>